Ankur Naskar

Gugan Thoppe, L. A. Prashanth, Ankur Naskar et al.

Reinforcement learning (RL) for exponential-utility optimization in discounted Markov decision processes (MDPs) lacks principled value-based algorithms. We address this gap in the fixed risk-aversion setting. Building on the Bellman-type equation for exponential utility studied in \cite{porteus1975optimality}, we derive two Q-value-style extensions and show that the associated operators are contractions in the $L_\infty$ and sup-log/Thompson metrics, respectively. We characterize their fixed points and prove that the induced greedy stationary policy is optimal for the exponential-utility objective among stationary policies. These structural results lead to two model-free algorithms: a two-timescale Q-learning--style algorithm, for which we establish almost-sure convergence and provide finite-time convergence rates via timescale separation, and a one-timescale algorithm governed by a sublinear power-law operator. Since the latter does not admit a global contraction in standard metrics, we prove its convergence using delicate arguments based on local Lipschitzness, monotonicity, homogeneity, and Dini derivatives, and provide a scalar finite-time analysis that highlights the challenges in obtaining convergence rates in the vector case. Our work provides a foundation for value-based RL under exponential-utility objectives.

Ankur Naskar, Gugan Thoppe, Vijay Gupta

Algorithms for solving \textit{nonlinear} fixed-point equations -- such as average-reward \textit{$Q$-learning} and \textit{TD-learning} -- often involve semi-norm contractions. Achieving parameter-free optimal convergence rates for these methods via Polyak--Ruppert averaging has remained elusive, largely due to the non-monotonicity of such semi-norms. We close this gap by (i.) recasting the averaged error as a linear recursion involving a nonlinear perturbation, and (ii.) taming the nonlinearity by coupling the semi-norm's contraction with the monotonicity of a suitably induced norm. Our main result yields the first parameter-free $\tilde{O}(1/\sqrt{t})$ optimal rates for $Q$-learning in both average-reward and exponentially discounted settings, where $t$ denotes the iteration index. The result applies within a broad framework that accommodates synchronous and asynchronous updates, single-agent and distributed deployments, and data streams obtained either from simulators or along Markovian trajectories.

Ankur Naskar, Gugan Thoppe, Utsav Negi et al.

Federated learning (FL) can dramatically speed up reinforcement learning by distributing exploration and training across multiple agents. It can guarantee an optimal convergence rate that scales linearly in the number of agents, i.e., a rate of $\tilde{O}(1/(NT)),$ where $T$ is the iteration index and $N$ is the number of agents. However, when the training samples arise from a Markov chain, existing results on TD learning achieving this rate require the algorithm to depend on unknown problem parameters. We close this gap by proposing a two-timescale Federated Temporal Difference (FTD) learning with Polyak-Ruppert averaging. Our method provably attains the optimal $\tilde{O}(1/NT)$ rate in both average-reward and discounted settings--offering a parameter-free FTD approach for Markovian data. Although our results are novel even in the single-agent setting, they apply to the more realistic and challenging scenario of FL with heterogeneous environments.